Do neuroimaging data have higher-order interactions?

Recent advances in the physics of complex systems have motivated the study of higher-order structure in neuroimaging data, including functional magnetic resonance imaging (fMRI), electroencephalography (EEG), and magnetoencephalography (MEG). Despite the proliferation of proposed measures, the meaning of "higher-order" is often left implicit, and the term is used inconsistently across studies. In this work, we clarify the distinction between higher-order interactions and higher-order dependencies. Higher-order interactions necessarily imply higher-order dependencies, but the converse does not hold: systems can exhibit higher-order dependencies in the absence of higher-order interactions, as in the case of Gaussian variables. We present several equivalent definitions of higher-order interactions for continuous data, illustrate them using simple model systems, and use them to assess the specificity of measures commonly applied in neuroimaging. We show that these measures are not specific to higher-order interactions. Consequently, although existing studies support the presence of higher-order dependencies in neuroimaging data, they do not provide conclusive evidence for the presence of higher-order interactions.